Explanation Given information: The f = x + 2 y + z subject to the constraints, x + 3 y + 2 z ≥ 40 x + y + z ≥ 30 x + y + z ≤ 100 Formula used: The mixed constraints by the use of Simplex method, (1) The problem is to maximize use f and to minimize use − f . (2) Make all the constraints to ≤ by multiply − 1 both the sides of the constraint and use slack variable to make simplex matrix. (3) Check the negative entry in the upper portion of the last column if entries are negative go to step 4 otherwise apply simplex method. (4) select the negative entry in the same row and take that column as pivot column. In pivot column find all quotients above the line for the entries then find the pivot from the smallest positive quotient. Then after completing the operation go to step 3. Calculation: Consider the provided function, f = 10 x + 30 y + 35 z And the constraints, x + 3 y + 2 z ≥ 40 x + y + z ≥ 30 x + y + z ≤ 100 Express all the constraints as ≤ constraints. Here, multiply x + 3 y + 2 z ≥ 40 , x + y + z ≥ 30 by − 1 to obtain ≤ constraints. So, the constraints are, − x − 3 y − 2 z ≤ 40 − x − y − z ≤ 30 x + y + z ≤ 100 Slack variables are used to write the inequalities as equations. The equations after adding slack variables s 1 , s 2 , s 3 are, − x − 3 y − 2 z + s 1 = − 40 − x − y − z + s 2 = − 30 x + y + z + s 3 = 100 The simplex matrix is, [ x y z s 1 s 2 s 3 − f − 1 − 3 − 2 1 0 0 0 − 40 − 1 − 1 − 1 0 1 0 0 − 30 1 1 1 0 0 1 0 100 1 2 1 0 0 0 1 0 ] The upper portion of last column has negative values. If values for the variables x , y , z , s 1 , s 2 , s 3 are computed then s 1 = − 40 , s 2 = − 30 is obtained. This violates the condition of simplex method that all the variables be non-negative. Choosing − 40 which is row one. In this row there are negative entries in column one, three and column two. Choosing the pivot column as column three. The pivot is determined from the quotients as, [ x y z s 1 s 2 s 3 − f − 1 − 3 − 2 1 0 0 0 − 40 − 1 − 1 − 1 0 1 0 0 − 30 1 1 1 0 0 1 0 100 1 2 1 0 0 0 1 0 ] − 40 / − 2 = 20 − 30 / − 1 = 30 100 / 1 = 100 The least positive quotient is 20 . Hence the pivot row is row one and pivot column is column three

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 4.5, Problem 24E

To determine

To calculate: The minimum value of f using simplex method for f=x+2y+z.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.5, Problem 23E

Chapter 4.5, Problem 25E

Students have asked these similar questions

2. In a computer network some pairs of computers are connected by network cables. Your goal is to set up the computers so that messages can be sent quickly from any computer to any other computer. For this you have identified each of the n com- puters uniquely with a number between 1 and n, and have decided that a message should consist of two such numbers, identifying the sender and the recipient, fol- lowed by the content of the message. As cables are relatively short, you can assume that sending a message across a single cable takes an amount of time that is the same irrespective of the length of the cable. You can further assume that at most one message travels between computer at any point, so that you don't have to worry about inference among messages. (a) Define a graph or network that models the computer network and allows you to answer the remaining parts of this question. (b) Consider two computers, a sender and a recipient. Using the graph or network you have defined,…

3. A spreadsheet consists of cells indexed by a row and a column. Each cell contains either a value or a formula that depends on the values of other cells. (a) Describe a graph, digraph, or network that models an arbitrary spreadsheet and allows you to answer the remaining parts of this question. (b) Explain, by referring to the graph, digraph, or network, when it is possible to change the value of cell x without changing the value of cell y. (c) Explain, by referring to the graph, digraph, or network, when it is possible to calculate the values of all cells in the spreadsheet. Consider the following spreadsheet with 5 rows, 7 columns, and 35 cells. For exam- ple, cell el contains a value, whereas cell al contains a formula that depends on the values cells el and 95. a b с 1 el+g5 al-c5 110 d al+cl 180 e f g f5-el c1+c2 2 al+b1 a2+c4 240 a2+c2 120 f5-e2 e3+e5 3 a2+b2 a3-c3 100 a3+c1 200 f5-e3 f1+f2 4 a3+b3 a4+c2 220 a4+c2 100 f5-e4 f3+f4 5 a4+b4 a5-c1 130 a5+c5 120 g3+g4 g1+g2 (d) Can…

1. Let W, U, and S be graphs defined as follows: • V(W) is the set of countries in the world; • V(U) is the set of countries in the European Union; V(S) is the set of countries in the Schengen Area; ● for X = {W,U,S}, E(X) is the set of pairs of countries in V(X) that share a land border. Recall that land borders between countries in the Schengen Area are special in that they can be crossed without a passport. (a) The notions of a country and a land border are somewhat ambiguous. Explain the notions you will use to get a precise definition of the graphs W, U, and S. (b) Is S a subgraph of U? Is U an induced subgraph of W? Justify your answers. (c) Using non-mathematical language, explain what it means for a country x if VEV(S) and dw (v) = 0. Give all such countries. Let A = {v Є V(W) \V(S) such that |Nw(v)| > 0 and Nw (v) ≤ V(S)}. (d) Using non-mathematical language, explain what the set A represents in terms of countries and land borders. Give a specific element of A or explain why A…

Chapter 4 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The minimum value of f using simplex method for f = x + 2 y + z .

Solution Summary: The author explains how to calculate the minimum value of f using simplex method.

Videos

To calculate: The minimum value of f using simplex method for f=x+2y+z.

Want to see the full answer?

Chapter 4 Solutions